The Complexity of Verifying Loop-Free Programs as Differentially Private

Authors Marco Gaboardi, Kobbi Nissim , David Purser

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Author Details

Marco Gaboardi
  • Boston University, MA, USA
Kobbi Nissim
  • Georgetown University, Washington, DC, USA
David Purser
  • University of Warwick, Coventry, UK
  • Max Planck Institute for Software Systems, Saarbrücken, Germany


Research partially done while M.G. and K.N. participated in the "Data Privacy: Foundations and Applications" program held at the Simons Institute, UC Berkeley in spring 2019.

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Marco Gaboardi, Kobbi Nissim, and David Purser. The Complexity of Verifying Loop-Free Programs as Differentially Private. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 129:1-129:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the problem of verifying differential privacy for loop-free programs with probabilistic choice. Programs in this class can be seen as randomized Boolean circuits, which we will use as a formal model to answer two different questions: first, deciding whether a program satisfies a prescribed level of privacy; second, approximating the privacy parameters a program realizes. We show that the problem of deciding whether a program satisfies ε-differential privacy is coNP^#P-complete. In fact, this is the case when either the input domain or the output range of the program is large. Further, we show that deciding whether a program is (ε,δ)-differentially private is coNP^#P-hard, and in coNP^#P for small output domains, but always in coNP^{#P^#P}. Finally, we show that the problem of approximating the level of differential privacy is both NP-hard and coNP-hard. These results complement previous results by Murtagh and Vadhan [Jack Murtagh and Salil P. Vadhan, 2016] showing that deciding the optimal composition of differentially private components is #P-complete, and that approximating the optimal composition of differentially private components is in P.

Subject Classification

ACM Subject Classification
  • Security and privacy
  • Theory of computation → Probabilistic computation
  • differential privacy
  • program verification
  • probabilistic programs


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